But cosmic rays don't hit each other head-on, they hit stationary nuclei, and the centre-of-momentum frame energy of these collisions is far lower than in the LHC.

I think your analysis ("scaremongering to make it sound more exciting") is probably correct, but I do worry a bit.

Actually cosmic rays with energies 1000 times that of the LHC hit about once per metre squared per year. I'm sure even hitting a fixed target this is higher CM energy. The main difference at the LHC would be the frequency of high energy collisions, so very very rare events would be more likely.
Still while scientists have pointed out possible dangers there are, as far as I am aware no 'anti-LHC' scientists campaigning against it. You'll always see more of this in the media than the scientific press.